Lemma 28.9.2. Let $X$ be a scheme. The following are equivalent:

1. $X$ is regular,

2. $X$ is locally Noetherian and all of its local rings are regular, and

3. $X$ is locally Noetherian and for any closed point $x \in X$ the local ring $\mathcal{O}_{X, x}$ is regular.

Proof. By the discussion in Algebra preceding Algebra, Definition 10.110.7 we know that the localization of a regular local ring is regular. The lemma follows by combining this with Lemma 28.5.2, with the existence of closed points on locally Noetherian schemes (Lemma 28.5.9), and the definitions. $\square$

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